3.996 \(\int \frac{(d+e x)^7}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=17 \[ \frac{(d+e x)^4}{4 c^2 e} \]

[Out]

(d + e*x)^4/(4*c^2*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.0173876, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(d+e x)^4}{4 c^2 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^7/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d + e*x)^4/(4*c^2*e)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.4063, size = 12, normalized size = 0.71 \[ \frac{\left (d + e x\right )^{4}}{4 c^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**7/(c*e**2*x**2+2*c*d*e*x+c*d**2)**2,x)

[Out]

(d + e*x)**4/(4*c**2*e)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00262482, size = 17, normalized size = 1. \[ \frac{(d+e x)^4}{4 c^2 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^7/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d + e*x)^4/(4*c^2*e)

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 16, normalized size = 0.9 \[{\frac{ \left ( ex+d \right ) ^{4}}{4\,{c}^{2}e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^7/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x)

[Out]

1/4*(e*x+d)^4/c^2/e

_______________________________________________________________________________________

Maxima [A]  time = 0.69576, size = 50, normalized size = 2.94 \[ \frac{e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^7/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="maxima")

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)/c^2

_______________________________________________________________________________________

Fricas [A]  time = 0.221024, size = 50, normalized size = 2.94 \[ \frac{e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^7/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="fricas")

[Out]

1/4*(e^3*x^4 + 4*d*e^2*x^3 + 6*d^2*e*x^2 + 4*d^3*x)/c^2

_______________________________________________________________________________________

Sympy [A]  time = 0.243851, size = 46, normalized size = 2.71 \[ \frac{d^{3} x}{c^{2}} + \frac{3 d^{2} e x^{2}}{2 c^{2}} + \frac{d e^{2} x^{3}}{c^{2}} + \frac{e^{3} x^{4}}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**7/(c*e**2*x**2+2*c*d*e*x+c*d**2)**2,x)

[Out]

d**3*x/c**2 + 3*d**2*e*x**2/(2*c**2) + d*e**2*x**3/c**2 + e**3*x**4/(4*c**2)

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^7/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError